To find the probability that a random sample has at least one member with the certain medical condition, we can use the concept of complementary probability.

Step 1: Define the problem

Let's denote:

• $p$ as the proportion of the community that has the medical condition.
• $n$ as the number of members in the sample.

We want to find the probability that at least one member in the sample has the medical condition.

Step 2: Calculate the probability of the complementary event

The complementary event is that none of the members in the sample have the medical condition.

The probability that a single member does not have the condition is $1 - p$.

Since the sample is random and each selection is independent, the probability that none of the $n$ members have the condition is: $(1 - p)^n$

Step 3: Calculate the desired probability

The probability that at least one member in the sample has the condition is the complement of the probability that none of the members have it. Thus: $\text{Probability} = 1 - (1 - p)^n$

Final Answer:

$\text{Probability} = 1 - (1 - p)^n$

This expression gives you the probability that at least one member in the sample has the medical condition.

Would you like further details on this problem, or have any questions? Here are some related questions:

1. How would you calculate the probability if the sample size $n$ is known?
2. What happens to the probability as $n$ increases?
3. How does a change in $p$ affect the probability of finding at least one member with the condition?
4. Can you explain the concept of complementary events with another example?
5. What is the expected number of people in the sample who have the condition?

Tip: Complementary probability is a powerful tool, especially when it's easier to calculate the probability of the event not happening.